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BERTRAND CURVE

Curve studied by Bertrand in 1850, Serret in 1851, Bioche
in 1889 and then Darboux.
Joseph Bertrand (1822-1900): French mathematician. [Gomes t2] p 447 , [Mir] p58 , [Berger] p 351, [Lelong- Ferrand] p 695, [Valiron] p 421, [Loria] p 90. |

Intrinsic equation: .
Cartesian parametrization: where is the parametrization of a curve with constant curvature (a skew circle), and is the parametrization of a curve with constant torsion. |

Bertrand curves are the 3D curves the curvature and torsion of which are linked by an affine non linear relation (hence the above intrinsic equation) - the linear case yields the helices.

A curve is a Bertrand curve if and only if there exists a curve different from with the same principal normal line as .

Except for the case of a circular helix, the curve is unique; the distance between two corresponding points along the common normal line is constant, and the angle formed by the corresponding tangents is constant.

Examples: the circular helix and, more generally, the
skew
circles (case where
*b* = 0; the angle formed by the tangents
is then a right angle and each curve is the locus of the centers of curvature
of the other).

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© Robert FERRÉOL 2018